On the existence of L
2
-valued thermodynamic entropy solutions
for a hyperbolic system with boundary conditions
Stefano Marchesani
Stefano Olla
May 13, 2019
Abstract
We prove existence of L2-weak solutions of a quasilinear wave equation with boundary conditions.
This describes the isothermal evolution of a one dimensional non-linear elastic material, attached to a fixed point on one side and subject to a force (tension) applied to the other side. The L2-valued
solutions appear naturally when studying the hydrodynamic limit from a microscopic dynamics of a chain of anharmonic springs connected to a thermal bath. The proof of the existence is done using a vanishing viscosity approximation with extra Neumann boundary conditions added. In this setting we obtain a uniform a priori estimate in L2, allowing us to use L2 Young measures, together with the classical tools of compensated compactness. We then prove that the viscous solutions converge to weak solutions of the quasilinear wave equation strongly in Lp, for any p ∈ [1, 2), that satisfy, in a weak sense, the boundary conditions. Furthermore, these solutions satisfy the Clausius inequality: the change of the free energy is bounded by the work done by the boundary tension. In this sense they are the correct thermodynamic solutions, and we conjecture their uniqueness.
Keywords: hyperbolic conservation laws, quasi-linear wave equation, boundary conditions, weak solutions, vanishing viscosity, compensated compactness, entropy solutions, Clausius inequality. Mathematics Subject Classification numbers: 35L40, 35D40
1
Introduction
The problem of existence of weak solutions for hyperbolic systems of conservation law in a bounded domain has been studied for solutions that are of bounded variation or in L∞ [7]. In the scalar
case some works extend to L∞ solutions, obtained from viscous approximations [20]. But viscous
approximations require extra boundary conditions, that are usually taken of Dirichlet type. We present here an approach based on viscosity approximations, where the extra boundary conditions are of Neumann type, to reflect the conservative nature of the viscous approximation. We consider here the quasilinear wave equation
(
rt− px= 0
pt− τ (r)x= 0
, (t, x) ∈ R+× [0, 1] (1.1)
where τ (r) is a strictly increasing regular function of r such that 0 < c1 ≤ τ′(r) ≤ c2, for some
constant c1, c2. In section 2 we will require some more technical assumption for τ . We add to the
system the following boundary conditions:
p(t, 0) = 0, τ (r(t, 1)) = ¯τ (t) (1.2)
and initial data
r(0, x) = r0(x), p(0, x) = p0(x). (1.3)
The boundary tension ¯τ : R+→ R is smooth and bounded with bounded derivative.
The equations (1.1) describe the isothermal evolution of an elastic material in Lagrangian co-ordinates. The material point x ∈ [0, 1] has a volume strain r(t, x) at time t (that can also have negative values), and momentum (velocity) p(t, x). The Eulerian position of the material point x, with respect to the position of the particle 0, is given by q(t, x) = R0xr(t, y)dy, so that we can identify the position of the material point x = 1 as the total extension of the material:
L(t) = q(t, 1) = Z1
0
r(t, y)dy. (1.4)
Let T < ∞ be given and arbitrary, and define QT := [0, T ] × [0, 1]. We shall construct weak
solutions ¯u(t, y) = (¯r(t, y), ¯p(t, y)) , (t, y) ∈ QT, to the quasilinear wave equation such that ¯u(t, ·) ∈
L2(0, 1) for all t ≤ T and satisfy the initial and boundary conditions in the following weak sense:
Z 1 0 ϕ(t, x)¯r(t, x)dx − Z 1 0 ϕ(0, x)r0(x)dx = Z t 0 Z 1 0 (ϕsr − ϕ¯ xp) dxds¯ (1.5) Z 1 0 ψ(t, x)¯p(t, x)dx − Z 1 0 ψ(0, x)p0(x)dx = Z t 0 Z 1 0 (ψsp − ψ¯ xτ (¯r)) dxds + Zt 0 ψ(s, 1)¯τ (s)ds (1.6)
for all functions ϕ, ψ ∈ C1(Q
T) such that ϕ(t, 1) = ψ(t, 0) = 0 for all t ≥ 0.
Define the free energy of the system, associated to a profile u(x) = (r(x), p(x)) ∈ L2(0, 1), as
F(u) := Z 1 0 p2(x) 2 + F (r(x)) dx (1.7)
where F (r) is a primitive of τ (r) (F′(r) = τ (r)), such that c1
2r
2≤ F (r) ≤ c2
2r
2for any r ∈ R. This
is possible thanks to the bounds we required on τ′.
The solution ¯u of (1.6) that we obtain has the following properties: • ¯u ∈ L∞(0, T ; L2(0, 1))
• ¯u(0, x) = u0(x) for a.e. x;
• For any φ ∈ C1([0, 1]), the application t 7→
Z 1 0
φ(x)¯u(t, x)dx (1.8)
is Lipschitz continuous over [0, T ]; • ¯u satisfies Clausius inequality:
F(¯u(t)) − F(u0) ≤ W (t), ∀t ∈ [0, T ] (1.9) where u0= (r0, p0) and W (t) := − Z t 0 ¯ τ′(s) Z 1 0 ¯ r(s, x)dxds + ¯τ (t) Z 1 0 ¯ r(t, x)dx − ¯τ(0) Z1 0 r0(x)dx (1.10)
is the work done by the external tension up to time t. In this sense we call our solution a ther-modynamic entropy solution. For general discussion of the connection of such therther-modynamic solutions to the usual definition of entropic solutions, see [11] and [4].
Remark. If ¯r(t, x) is differentiable with respect to time, we may perform an integration by parts and obtain W (t) = Z t 0 ¯ τ (s)dL(s). (1.11)
The construction of the solution is obtained from the following viscosity approximation ( rtδ− pδx= δrδxx pδ t− τ (rδ)x= δpδxx , (t, x) ∈ R+× [0, 1] (1.12)
with boundary conditions
pδ(t, 0) = 0, τ (rδ(t, 1)) = ¯τ (t), pδx(t, 1) = 0, rxδ(t, 0) = 0 (1.13)
and initial data
rδ(0, x) = rδ0(x), pδ(0, x) = pδ0(x) (1.14)
such that rδ
0 and pδ0 are compatible with the boundary conditions, regular enough (see (3.4) and
(3.5)) and converge to r0 and p0, respectively, as δ → 0.
Note that in the viscous approximation we have added two Neumann boundary conditions, that reflect the conservative nature of the viscous perturbation. Under these conditions we have
Z 1 0 |uδ(t, x)|2dx + δ Z t 0 Z1 0 |uδx(s, x)|2dxds ≤ C, ∀t ≥ 0 (1.15)
where C is independent of t and δ. It is thus clear that {uδ}
δ>0 and {
√ δuδ
x}δ>0 are uniformly
bounded in L2(Q
T). Then we rely on the existence of a family of bounded Lax entropy-entropy
fluxes as in [23], [21] and [22], that allows us to apply the compensated compactness in the L2
version. The conditions assumed on τ (r) are in fact those required to apply [23] results. Under a slight different set of conditions, another Lp extension of the compensated compactness argument can be found in [13].
1.1
Physical motivations
The problem arises naturally considering hydrodynamic limit for a non-linear chain of anharmonic oscillators in contact with a heat bath at a given temperature [16, 15]. This microscopic dynamics models an isothermal transformation with two locally conserved quantities that evolve, on the macroscopic scale, following (1.1).
Consider N + 1 particles on the real line and, for i = 0, . . . N , call qiand pithe positions and the
momenta of the i-th particle, respectively. Particles i and i − 1 interact via a nonlinear potential V (qi− qi−1). Particle i = 0 is at position q0= 0 and does not move, i.e. p0(t) = 0. There is a time
dependent force (tension) ¯τ (t) acting on the last particle. Then, defining ri:= qi− qi−1 we have a
system with Hamiltonian
HN(t) = N X i=1 p2 i 2 + V (ri) − ¯τ(t) N X i=1 ri. (1.16)
The interaction with a heat bath at temperature β−1 is modeled by a stochastic perturbation of the dynamics, that acts as a microscopic stochastic viscosity. Defining the discrete gradient and laplacian as
∇ai= ai+1− ai, ∆ai= ai+1+ ai−1− 2ai,
the evolution equations are then given by the following system of stochastic differential equations: dr1= p1dt + δ∇V′(r1)dt − p 2β−1δ d ew 1 dri= ∇pi−1dt + δ∆V′(ri)dt − p 2β−1δ ∇d ew i−1, 2 ≤ i ≤ N − 1 drN= ∇pN −1dt + δ (¯τ (t) + V′(rN −1) − 2V′(rN)) − p 2β−1δ ∇d ewN −1, dp1= ∇V′(r1)dt + δ (p2− 2p1) dt − p 2β−1δ ∇dw1, dpj= ∇V′(rj)dt + δ∆pjdt − p 2β−1δ ∇dw j−1, 2 ≤ j ≤ N − 1 dpN= (¯τ (t) − V′(rN))dt − δ∇pN −1dt + p 2β−1δ dwN −1 (1.17)
Here β−1> 0 is the temperature of the heat bath, and {wi}N −1i=1 , { ewi}N −1i=1 are families of independent
Brownian motions. The parameter δ is the intensity of the action of the heat bath, and is chosen depending on N such that δ ∼ o(N). When δ = 0, equations (1.17) are just the Newton deterministic equations for the Hamiltonian (1.16). Notice the correspondence of the boundary conditions in (1.17) with the one chosen in (1.12).
One of the effects of the action of the stochastic heat bath is to fix, in a large time scale, the variance of the velocities (i.e. the temperature) at β−1, and establish a local equilibrium, where
space-time averages of V′(r
i) around a macroscopic particle number [N x] at a macroscopic time
N t converges to the equilibrium tension τ (r(t, x), β) at temperature β−1and volume stretch r(t, x).
Since β is fixed by the heat bath and do not evolve in time, we drop it from the notation in the sequel.
The hydrodynamic limit consists in proving that, for any continuous function G(x) on [0, 1],
1 N N X i=1 G i N ri(N t) pi(N t) −→ N →∞ Z 1 0 G(x) r(t, x) p(t, x) dx, (1.18)
in probability, with (r(t, x), p(t, x)) satisfying (1.5), (1.6). Of course a complete proof would require the uniqueness of such L2 valued solutions that satisfy (1.9): this remains an open problem. The
results contained in [15] states that the limit distribution of the empirical distribution defined on the RHS of (1.18), concentrates on the possible solutions of (1.5) and (1.6) that satisfy (1.9). Since we have no uniqueness result, we cannot assure that the solutions constructed in the present paper coincide with those obtained with the hydrodynamic limit from (1.17). One can however conjecture that this is the case.
This stochastic model was already considered by Fritz [12] in the infinite volume without bound-ary conditions, and in [16], but without the characterisation of the boundbound-ary conditions.
In the hydrodynamic limit only L2 bounds are available and we are constrained to consider L2 valued solutions. Since these solutions do not have definite values on the boundary, boundary
conditions have only a dynamical meaning in the sense of an evolution in L2given by (1.5), (1.6).
2
Hyperbolic system and the existence of weak solutions
For r, p : R+× [0, 1] → R, consider the hyperbolic system
( rt− px= 0 pt− τ (r)x= 0 , p(t, 0) = 0 r(t, 1) = τ −1(¯τ (t)) p(0, x) = p0(x) r(0, x) = r0(x) (2.1)
The nonlinearity τ ∈ C3(R) is chosen to have the following properties.
(τ -i) c1≤ τ′(r) ≤ c2 for some c1, c2> 0 and all r ∈ R;
(τ -ii) τ′′
(r) 6= 0 for all r ∈ R; (τ -iii) τ′′(r), τ′′′
(r) ∈ L2(R) ∩ L∞
(R).
We also assume that ¯τ : R+→ R is smooth. Moreover, there is a time T⋆such that ¯τ′(t) = 0 for all
t ≥ T⋆. The initial data r0, p0∈ L2(0, 1) are compatible with the boundary conditions.
Remark. Conditions (τ -i) and (τ -ii) ensure that the system is strictly hyperbolic and genuinely nonlinear, respectively. Condition (τ -iii) is used later on to ensure some boundedness properties of the Lax entropies.
Theorem 2.1. System (2.1) admits a weak solution ¯u = (¯r, ¯p) in the sense of (1.5) and (1.6), such that ¯u ∈ L∞(0, T ; L2(0, 1)), ¯u(0, x) = u
0(x) for a.e. x; and it satisfies the Clausius inequality:
F(¯u(t)) − F(u0) ≤ W (t), ∀t ∈ [0, T ] (2.2)
3
Viscous approximation and energy estimates
We consider the following parabolic approximation of the hyperbolic system (2.1) ( rδ t − pδx= δrδxx pδ t− τ (rδ)x= δpδxx , (t, x) ∈ R+× [0, 1] (3.1)
for δ > 0, with the boundary conditions:
pδ(t, 0) = 0, rδ(t, 1) = τ−1(¯τ (t)), pδx(t, 1) = 0, rδx(t, 0) = 0, (3.2)
and initial data:
pδ(0, x) = pδ0(x), rδ(0, x) = rδ0(x). (3.3)
The initial data rδ
0, pδ0 ∈ C∞([0, 1]) are mollifications of r0 and p0 compatible with the boundary
conditions:
pδ0(0) = 0 rδ0(1) = τ−1(¯τ (0)), ∂xpδ0(1) = 0 ∂xrδ0(0) = 0. (3.4)
Moreover, there is C independent of δ such that
krδ0kL2+ kpδ0kL2+ k √ δ∂xr0δkL2. + k √ δ∂xpδ0kL2≤ C (3.5) and (rδ 0, pδ0) → (r0, p0) strongly in L2(0, 1).
As shown in [2] in a more general setting, this system admits a global classical solution (rδ, pδ),
with
rδ, pδ∈ C1(R+; C0([0, 1])) ∩ C0(R+; C2([0, 1])).
Remark. (i) We added two extra Neumann conditions, namely pδ
x(t, 1) = rδx(t, 0) = 0. These
conditions reflect the conservative nature of the viscous perturbation, and are required in order to obtain the correct production of free energy.
(ii) One could introduce a nonlinear viscosity term: δτ (rδ)
xx in place of δrxxδ . This is a term
which comes naturally from a microscopic derivation of system (3.1), as described in the introduction (see also [14]). Nevertheless, this does not drastically change the problem, thus we shall consider only the linear viscosity δrδxx.
Theorem 3.1(Energy estimate). There there is a constant C > 0 independent of t and δ such that Z 1 0 |uδ(t, x)|2dx + δ Z t 0 Z 1 0 |uδx(s, x)|2dxds ≤ C (3.6)
for all t ≥ 0 and δ > 0.
Proof. Let F be a primitive of τ such that c1 2r
2
≤ F (r) ≤ c22r2. By a direct calculation we have Z 1 0 (pδ)2 2 + F (r δ) dx t=T t=0 + Z T 0 Z 1 0 δ(rδx)2+ δ(pδx)2 dxdt = Z T 0 ¯ τ (t) Z 1 0 rδtdxdt (3.7) = ¯ τ (t) Z 1 0 rδdx t=T t=0 − Z T 0 ¯ τ′(t) Z1 0 rδdxdt. (3.8)
Write, for some ε > 0 to be chosen later,
¯ τ (T ) Z 1 0 rδ(T, x)dx ≤ |¯τ(T )| 1 2ε+ ε 2 Z 1 0 (rδ)2(T, x)dx ≤C2ε¯τ +Cτ¯ε 2 Z1 0 (rδ)2(T, x)dx (3.9) where Cτ¯= supt≥0(|¯τ(t)| + |¯τ ′ (t)|) depends on ¯τ only.
Using F (r) ≥c21r2 we obtain c1 2 − Cτ¯ε 2 Z 1 0 (rδ)2(T, x)dx +1 2 Z 1 0 (pδ)2(T, x)dx + Z T 0 Z 1 0 δ(rδx)2+ δ(pδx)2 dxdt ≤ Cτ¯ 2ε + C0− Z T 0 ¯ τ′(t) Z 1 0 rδ(t, x)dxdt. (3.10)
Recall that there is T⋆> 0 such that ¯τ′(t) = 0 for t ≥ T⋆. Then, for T < T⋆, we write
c1 2 − Cτ¯ε 2 Z 1 0 (rδ)2(T, x)dx +1 2 Z 1 0 (pδ)2(T, x)dx + Z T 0 Z 1 0 δ(rδx)2+ δ(pδx)2 dxdt ≤ C2ε¯τ + C0+ C2 ¯ τ 2 T + 1 2 ZT 0 Z1 0 (rδ)2(t, x)dx (3.11) ≤C2ετ¯ + C0+C 2 ¯ τ 2 T + 1 2 Z T 0 Z 1 0 (rδ)2(t, x) + (pδ)2(t, x)dxdt (3.12) +1 2 Z T 0 Z t 0 Z 1 0 δ(rδx)2+ δ(pδx)2 dxdsdt
where C0 depends on the initial data only. Choosing ε = c1/(2Cτ¯) gives
c1 4J(T ) ≤ C0+ C2¯τ c1 +C 2 ¯ τ 2 T + 1 2 Z T 0 J(t)dt, (3.13) where J(t) = Z 1 0 (rδ)2(t, x) + (pδ)2(t, x)dx + Z t 0 Z 1 0 δ(rxδ)2+ δ(pδx)2 dxds. (3.14)
We apply Gronwall’s inequality. This, together with T < T⋆, gives
J(T ) ≤4c1C0+ 2C 2 ¯ τ(2 + c1T ) c2 1 exp 2T c1 ≤4c1C0+ 2C 2 ¯ τ(2 + c1T⋆) c2 1 exp 2T⋆ c1 := C0(c1, ¯τ ), (3.15)
for all T ∈ [0, T⋆), where C0(c1, ¯τ ) is independent of T and δ
On the other hand, if T ≥ T⋆, we have
c1 2 − Cτ¯ε 2 Z 1 0 (rδ)2(T, x)dx +1 2 Z 1 0 (pδ)2(T, x)dx + Z T 0 Z 1 0 δ(rδx)2+ δ(pδx)2 dxdt ≤ C2ε¯τ + C0− Z T⋆ 0 ¯ τ′(t) Z 1 0 rδ(t, x)dxdt (3.16)
and the integral at the right-hand side is uniformly bounded in T , δ and δ, since
− ZT⋆ 0 ¯ τ′(t) Z 1 0 rδ(t, x)dxdt ≤ Cτ¯ ZT⋆ 0 Z 1 0 |rδ(t, x)|dxdt ≤ Cτ¯T⋆ 1 T⋆ Z T⋆ 0 Z 1 0 (rδ)2(t, x)dxdt 1/2 ≤ Cτ¯ √ T⋆ Z T⋆ 0 J(t)dt 1/2 ≤ Cτ¯T⋆ p C0(c1, ¯τ ) (3.17)
From (3.7) we also immediately obtain the following Corollary 3.2 (Viscous Clausius inequality).
F(uδ(t)) − F(uδ0) ≤ − Z t 0 ¯ τ′(s) Z 1 0 rδ(s, x)dx + ¯τ (t) Z 1 0 rδ(t, x)dx − ¯τ(0) Z1 0 r0δ(x)dx. (3.18)
4
L
2Young measures and compensated compactness
Throughout this section, for any fixed T > 0 let uδ(t, x) := (rδ(t, x), pδ(t, x)) be a strong solution
of (3.1) on QT. By Theorem 3.1 and after a time integration over [0, T ] we obtain
kuδkL2(QT)≤ C (4.1)
for some C independent of δ. Thus we can extract from {uδ}
δ>0 a subsequence that is weakly
convergent in L2(Q
T). Namely, up to a subsequence, there exists ¯u = (¯r, ¯p) ∈ L2(QT) such that
lim δ→0 Z QT uδϕ = Z QT ¯ uϕ, ∀ϕ ∈ L2(QT). (4.2)
All the limits δ → 0 taken below are intended along a chosen subsequence. In this section we want to show that for any φ ∈ L2(QT) we have
lim δ→0 Z QT φ(t, x)τ (rδ(t, x)) dx dt = Z QT φ(t, x)τ (¯r(t, x)) dx dt. (4.3)
This is done using a L2version of the compensated compactness, which is usually performed in L∞.
From the solution uδ(t, x), we define the following Young measure on Q T× R2:
νt,xδ := δuδ(t,x), (4.4)
which is a Dirac mass centred at uδ, i.e.
Z QT J(t, x)f (uδ(t, x)) dx dt = Z QT Z R2 J(t, x)f (ξ)dνt,xδ (ξ)dxdt
for all mesurable J : QT → R and f : R2→ R.
Since we have L2 bounds on uδ, we refer at νδ
t,x as a L2-Young measure [3]. In particular we
have, from (4.1) Z QT Z R2|ξ| 2 dνt,xδ (ξ) dx dt ≤ C. (4.5)
We call Y the set of Young measures on QT× R2 and we make it a metric space by endowing it
with the Prohorov’s metric. By proposition 4.1 of [5], the set
KC:= ν ∈ Y : Z QT Z R2|ξ| 2dν t,x(ξ)dxdt ≤ C (4.6)
is compact in Y. Then, by the fundamental theorem for Young measures ([3], section 2), there exists ¯ νt,x∈ Y so that, up to a subsequence, lim δ→0 Z QT Z R2 J(t, x)f (ξ)dνt,xδ (ξ)dxdt = Z QT Z R2 J(t, x)f (ξ)d¯νt,x(ξ)dxdt (4.7)
for all continuous and bounded J : QT → R and f : R2 → R. We shall simply write νδ → ¯ν in
place of (4.7). By a simple adaptation of proposition 4.2 of [5], (4.7) can be extended to a function f : R2→ R such that f(ξ)/|ξ|2→ 0 as |ξ| → +∞.
In order to obtain (4.3), we need to prove that the limit Young measure ¯ν is a Dirac mass: ¯
νt,x = δu(t,x)¯ , for some ¯u ∈ L2(QT) and for almost every (t, x) ∈ QT. This is done using the
Definition 4.1. A Lax entropy-entropy flux pair for system (2.1) is a couple of differentiable func-tions (η, q) : R2→ R2 such that
(
ηr+ qp= 0
τ′(r)η
p+ qr= 0
. (4.8)
We show that Tartar’s equation holds for any two suitable entropy pairs (η, q) and (η′, q′) to be
specified below and almost all (t, x) ∈ QT:
hηq′− η′q, ¯νt,xi = hη, ¯νt,xihq′, ¯νt,xi − hη′, ¯νt,xihq, ¯νt,xi, (4.9) where hf, ¯νt,xi := Z R2 f (ξ)d¯νt,x(ξ) (4.10)
for any measurable f . We employ the following argument due to Shearer [23].
Accordingly to Lemma 2 in [23], there exists a family of half-plain supported entropy-entropy fluxes (η, q) such that η and q are bounded together with their first and second derivatives. These are explicitly given as follows. We define z(r) :=R0rpτ′(ρ)dρ and we define the Riemann coordinates
w1= p + z, w2= p − z. We also pass from the dependent variables η, q to H,Q as follows:
η = 1 2(τ ′ )−1/4(H + Q) (4.11) q = 1 2(τ ′ )+1/4(H − Q) (4.12) so that (4.8) becomes ( Hw1= aQ Hw2= −aQ , (4.13) where a(w1− w2) = τ′′r w1− w2 2 8τ′r w1− w2 2 3/2. (4.14)
Then we fix ¯w1, ¯w2 ∈ R and we solve (4.13) with Goursat data given on the lines w1 = ¯w1 and
w2= ¯w2:
H( ¯w1, w2) = g(w2)
Q(w1, ¯w2) = 0,
(4.15)
where g is continuous and compactly supported. Then one can explicitly solve (4.13) and get
H(w1, w2) = g(w2) + ∞ X n=1 (Ang)(w1, w2) Q(w1, w2) = − Z w2 ¯ w2 a(w1− v)H(w1, v)dv, (4.16)
where the operator A acts on functions f ∈ L1
loc(R2) as follows: (Af)(w1, w2) = − Z w1 ¯ w1 Z w2 ¯ w2
a(v − w2)a(v − u)f(v, u)dudv. (4.17)
Finally, going back to η and q and using our assumptions on τ it is possible to show ([23], Lemma 2) that η and q are bounded, together with their first and second derivatives.
Now we have a suitable family of entropy-entropy flux pair, we use Tartar-Murat Lemma in order to derive Tartar’s equation (4.9). We evaluate (η, q) along the approximate solutions uδ and
compute the entropy production:
η(uδ)t+ q(uδ)x= δ ηrrδx+ ηppδx x− δ ηrr(rδx)2+ ηpp(pxδ)2+ 2ηrprδxpδx (4.18)
Since ηr and ηpare bounded and
√ δrδx,
√
δpδx are bounded in L2(QT), we have
lim δ→0δ ηrrδx+ ηppδx x= 0 in H −1 (QT), (4.19) while δ ηrr(rxδ)2+ ηpp(pxδ)2+ 2ηrprδxpδx L1(Q T) ≤ C (4.20) uniformly with respect to δ. Thus we have have obtained an equality of the form
η(uδ)t+ q(uδ)x= χδ+ ψδ,
where {χδ}
δ>0 lies in a compact set of H−1(QT) and {ψδ}δ>0 is bounded in L1(QT). Moreover,
since η and q are bounded, {η(uδ)t+ q(uδ)x}δ>0 is bounded in W−1,p(QT) for some p > 2.
Therefore, we can apply Tartar-Murat and the div-curl lemma (cf [9], Theorem 16.2.1 and Lemma 16.2.2) and obtain Tartar’s equation (4.9).
The final step is to use Tartar’s equation to prove that the support of the limit Young measure ¯
νt,xis a point. This is done in lemmas 4 to 7 of [23] and leads to the following
Proposition 4.2. There exists a ¯u ∈ L2(Q
T) such that ¯νt,x = δu(t,x)¯ for almost all (t, x) ∈ QT.
Moreover, uδ→ ¯u strongly in Lp(Q
T) for any p ∈ [1, 2).
4.1
Regularity
Proposition 4.3. For the function ¯u obtained in section 4,
¯
u ∈ L∞(0, T ; L2(0, 1)).
Proof. Since uδ→ ¯u in Lpstrong for p < 2, we can extract a subsequence {uδk
}k∈N that converges
pointwise to ¯u for almost all t and x. In particular, for almost all t, the sequence uδk(t, x) converges
for almost all x. Therefore, by Fatou lemma and Theorem 3.1, Z 1 0 |¯u(t, x)| 2 dx ≤ lim infk→∞ Z 1 0 |u δk (t, x)|2dx ≤ C (4.21) for almost all t ∈ [0, T ].
The proof is of next lemma is standard and therefore omitted.
Lemma 4.4. Let a(t) := τ−1(¯τ (t)). Then, the solutions (rδ, pδ) of the viscous system (3.1) can be
written as follows: rδ(t, x) = a(t) + Z 1 0 Gδr(x, x′, t)(rδ0(x′) − a(0))dx′+ (4.22) + Z t 0 Z 1 0 Gδr(x, x′, t − t′)(∂x′pδ(t′, x′) − a(t′))dx′dt′ pδ(t, x) = Z 1 0 Gδp(x, x ′ , t)pδ0(x ′ )dx′+ Z t 0 Z 1 0 Gδp(x, x ′ , t − t′)∂x′τ (rδ(t′, x′))dx′dt′ (4.23) where the Gδ
r and Gδpare Green functions of the heat operator ∂t− δ∂xxwith homogeneous boundary
conditions: Gδr(1, x ′ , t) = ∂xGδr(0, x ′ , t) = 0 (4.24) Gδp(0, x′, t) = ∂xGδp(1, x′, t) = 0 (4.25) for all x′ ∈ [0, 1], t ≥ 0 and δ > 0.
The Green’s functions Gδr(x, x′, t) and Gδp(x, x′, t) are symmetric under the exchange of x and
x′. Moreover we have the following identities
∂xGδp(x, x ′ , t) = −∂x′Gδr(x, x ′ , t), (4.26) ∂xGδr(x, x′, t) = −∂x′Gδp(x, x′, t). (4.27)
Finally, Gδr and Gδpare subprobabilty densities, meaning
0 ≤ Z 1 0 Gδr(x, x′, t)dx′≤ 1 (4.28) 0 ≤ Z 1 0 Gδp(x, x′, t)dx′≤ 1 (4.29)
for any x, t and δ.
Remark. Although we will not use them, explicit expressions are available for the Gδ
r and Gδp, namely Gδp(x, x′, t) = X n odd e−tδλn sin√λnx sin√λnx′ (4.30) Gδr(x, x ′ , t) = X n odd e−tδλn cos√λnx cos√λnx′ , (4.31) with λn= n2π2 4 .
Proposition 4.5. For any φ ∈ C1([0, 1]), the application t 7→ Iφ(t) :=
Z 1 0
φ(x)¯u(t, x)dx (4.32)
is Lipschitz continuous. Consequently ¯u(t, ·) ∈ L2(0, 1) for all t ≥ 0.
Proof. We prove the statement for ¯p, as the proof for ¯r is similar. Furthermore, we prove the proposition only between 0 and t, as in the general case, say between t1 and t, it is enough to
replace the initial term pδ
0(x) with pδ(t1, x). We let Iφδ(t) := Z 1 0 φ(x)pδ(t, x)dx (4.33) and evaluate Iφ(t) − Iφ(0) = Z 1 0 Z 1 0 φ(x)Gδp(x, x′, t)pδ0(x′)dx′dx − Z1 0 φ(x)pδ0(x)dx+ (4.34) + Z t 0 Z 1 0 Z 1 0 φ(x)Gδp(x, x′, t − t′)∂x′τ (rδ(t′, x′))dxdx′dt′ = Z 1 0 φ(x) Z1 0 h Gδp(x, x ′ , t)pδ0(x ′ ) − pδ0(x) i dx′dx+ (4.35) + Z t 0 Z 1 0 Z 1 0 φ(x)∂xGδr(x, x′, t − t′)τ (rδ(t′, x′))dxdx′dt′+ Z t 0 Z1 0 φ(x)Gδp(x, 1, t − t′)¯τ (t′)dxdt′,
where we have used the symmetry of Gδ
pas well as the property ∂x′Gδ
p = −∂xGδr. The boundary term is estimated as Zt 0 Z 1 0 φ(x)Gδp(x, 1, t − t ′ )¯τ (t′)dxdt′ ≤ Cτ¯ Zt 0 Z 1 0 φ(x)Gδp(x, 1, t − t ′ )dx dt ≤ tCτ¯kφkL2, (4.36)
by Jensen’s inequality and since Gδpis a subprobability density.
We estimate the term involving τ as Z t 0 Z 1 0 Z1 0 φ(x)∂xGδr(x, x′, t − t′)τ (rδ(t′, x′))dxdx′dt′ = Z t 0 Z 1 0 Z 1 0 φ′(x)Gδ r(x, x′, t − t′)τ (rδ(t′, x′))dxdx′dt′ + (4.37) + φ(0) Z t 0 Z 1 0 Gδr(0, x′, t − t′)τ (rδ(t′, x′))dx′dt′ ≤ tkφ′kL2 1 t Z t 0 Z 1 0 Gδp(·, x′, t − t′)τ (rδ(x′, t′))dx′dt′ L2 + (4.38) + tkφkL∞ 1 t Z t 0 Z 1 0 Gδr(0, x′, t − t′)τ (rδ(t′, x′))dx′dt′ ≤ t kφ′kL2+ kφkL∞ 1 t Z t 0 Z 1 0 τ (rδ(x′, t′))2dx′dt′ 1/2 ≤ Ct (4.39) where C is independent of t and δ.
In order to estimate the first term of (4.35) we write Z 1 0 h Gδp(x, x′, t)pδ0(x′) − pδ0(x) i dx′=etδ∂xx − 1pδ0(x) (4.40) = Z tδ 0 ∂xxes∂xxpδ0(x)ds. (4.41) Therefore, we obtain Z 1 0 Z 1 0 h Gδp(x, x ′ , t)pδ0(x ′ ) − pδ0(x) i dx′dx ≤ Ztδ 0 Z 1 0 φ(x)∂xxes∂xxpδ0(x)dx ds (4.42) = Ztδ 0 Z 1 0 φ′(x)∂xes∂ xx pδ0(x)dx ds (4.43) ≤ kφ′ kL2 Z tδ 0 ∂xes∂xxpδ0 L2ds. (4.44)
Finally, since, by standard results on the heat equation, the application
s 7→ ∂xes∂xxpδ0 L2 (4.45) is decreasing, we have Z tδ 0 ∂xes∂xxpδ0 L2ds ≤ t √ δk√δ∂xpδ0kL2 ≤ Ct, (4.46)
where we have used the assumption that {√δ∂xpδ0}δ>0is bounded in L2(0, 1). We have also assumed,
without loss of generality, δ ≤ 1.
Putting everything together, we have obtained Iφδ(t) − Iφδ(0) ≤ Ct (4.47)
for some constant C independent of t and δ. This leads to the conclusion after passing to the limit δ → 0.
5
Proof of theorem 2.1 and Clausius inequality
All is left to prove is that the function ¯u obtained in the previous section is to a weak solution of the hyperbolic system (2.1), in the sense of Section 2. Let ψ ∈ C1(QT) with ψ(t, 0) = 0 for all t ∈ [0, T ].
Then, for any t ∈ [0, T ] we have 0 = Z t 0 Z 1 0 ψpδs− ψτ(rδ)x− δψpδxx dxds = Z 1 0 ψ(t, x)pδ(t, x)dx − Z 1 0 ψ(0, x)pδ0(x)dx+ (5.1) − Zt 0 Z 1 0 ψspδ− ψxτ (rδ) − δψxpδx dxds − Z ∞ 0 ψ(t, 1)¯τ (t)dt.
where we have used the initial-boundary conditions τ (rδ(t, 1)) = ¯τ (t) and pδ
x(t, 1) = 0, pδ(0, x) =
pδ
0(x) as well as ψ(t, 0) = 0. Since pδ0 converges to p0 in L2(0, 1), we have
lim δ→0 Z 1 0 ψ(0, x)pδ0(x)dx = Z 1 0 ψ(0, x)p0(x)dx. (5.2)
Furthermore, Theorem 3.1 implies√δpδx∈ L2(QT), consequently
Rt 0 R1 0 δψxp δ xdxds vanishes as δ →
0. Moreover, (4.2) implies, along the subsequence that defines ¯u = (¯r, ¯p),
lim δ→0 Z t 0 Z 1 0 ψtpδdxds = Z t 0 Z 1 0 ψspdxdt,¯ (5.3)
while by (4.3) we have that
lim δ→0 Z t 0 Z 1 0 ψxτ (rδ)dxds = Zt 0 Z 1 0 ψxτ (¯r)dxds,
so that (1.6) is satisfied. The (1.5) is linear and it follows similarly. Proposition 5.1. The solution ¯u satisfies Clausius inequality
F(¯u(t)) − F(u0) ≤ W (t) (5.4)
for all t ∈ [0, T ], where W (t) = − Z t 0 ¯ τ′(s) Z 1 0 ¯ r(s, x)dx + ¯τ (t) Z 1 0 ¯ r(t, x)dx − ¯τ(0) Z 1 0 r0(x)dx. (5.5)
Proof. By Proposition 4.3, Corollary 3.1, and Lemma 4.5, we have, for all t ∈ [0, T ], Z1 0 ¯ p2(t, x) 2 + F (¯r(t, x)) dx ≤ lim inf k→∞ Z 1 0 (pδk)2(t, x) 2 + F (r δk(t, x)) dx (5.6) ≤ lim k→∞ F(uδk 0 ) − Z t 0 ¯ τ′(s) Z1 0 rδk (s, x)dxds + ¯τ (t) Z 1 0 rδk (t, x)dx − ¯τ(0) Z 1 0 rδk 0 (x)dx (5.7) = F(u0) − Zt 0 ¯ τ′(s) Z 1 0 ¯ r(s, x)dxds + ¯τ (t) Z 1 0 ¯ r(t, x)dx − ¯τ(0) Z 1 0 r0(x)dx, (5.8)
where we have used the fact that uδ
0 converges to u0 in L2 strongly in order to conclude that
F(uδ0) → F(u0). Moreover, all the integrals are well defined, since the application
t 7→ Z1 0 ¯ r(t, x)dx is continuous.
Thanks to the Clausius inequality, the solutions we have constructed are natural candidates for being the thermodynamic entropy solution of the equation (3.1) and one can conjecture that such limit is unique.
Acknowledgments
This work has been partially supported by the grants ANR-15-CE40-0020-01 LSD of the French National Research Agency. We thank Olivier Glass for very helpful discussions. We also thank an anonomous referee, whose critics and comments helped us to improve the first version of this article.
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CEREMADE, UMR-CNRS, Universit´e de Paris Dauphine, PSL Research University
Place du Mar´echal De Lattre De Tassigny, 75016 Paris, France olla@ceremade.dauphine.fr
Stefano Marchesani GSSI,
Viale F. Crispi 7, 67100 L’Aquila, Italy stefano.marchesani@gssi.it
